The Fermi-Hubbard model only contains nearest-neighbor interactions and, according to [1]}, can be near optimally simulated with \(\mathcal {O}\left(n^{1+1/p}\right)\)  gates. On the other hand, recent work [2]} shows that Trotterization algorithm has gate complexity \(\mathcal {O}\left(n\eta ^{1+1/p}\right)\)  when restricted to the \(\eta \) -electron manifold. Our result improves over those previous work by combining the sparsity of interactions, commutativity of the Hamiltonian and information about the initial state.
